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Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Cyclicity of CM elliptic curves modulo p
 TRANSACTIONS OF AMERICAN MATHEMATICAL SOCIETY
, 2003
"... Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p ≤ x for which E(Fp) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.P. ..."
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Cited by 5 (1 self)
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Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p ≤ x for which E(Fp) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula, and also to provide explicit error terms in the formula.
Reductions of an elliptic curve and their TateShafarevich groups
, 2004
"... Abstract. In this paper we study the TateShafarevich groups IIIp of the reductions modulo primes p of an elliptic curve E/Q considered as being defined over their function fields. Assuming GRH when E has no CM, we show that IIIp is trivial for a positive proportion of primes p, provided E has an ir ..."
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Abstract. In this paper we study the TateShafarevich groups IIIp of the reductions modulo primes p of an elliptic curve E/Q considered as being defined over their function fields. Assuming GRH when E has no CM, we show that IIIp is trivial for a positive proportion of primes p, provided E has an irrational point of order two. 1.
Uniform results for Serre’s theorem for elliptic curves
, 2005
"... A celebrated theorem of Serre from 1972 asserts that if E is an elliptic curve defined over Q and without complex multiplication, then its associated mod ℓ representation is surjective for all sufficiently large primes ℓ. In this paper we address the question of what sufficiently large means in Serr ..."
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A celebrated theorem of Serre from 1972 asserts that if E is an elliptic curve defined over Q and without complex multiplication, then its associated mod ℓ representation is surjective for all sufficiently large primes ℓ. In this paper we address the question of what sufficiently large means in Serre’s theorem. More precisely, we obtain a uniform version of Serre’s theorem for nonconstant elliptic curves defined over function fields, and a uniform version of Serre’s theorem for oneparameter families of elliptic curves defined over Q.